Greeks initially beat back a bloodsucker (4)
I believe the answer is:
gnat
'a bloodsucker' is the definition.
(I've seen this before)
'greeks initially beat back' is the wordplay.
'initially' indicates taking the first letters.
'beat' becomes 'tan' (I've seen this before**).
'back' says the letters should be written in reverse.
The first letter of 'greeks' is 'g'.
'tan' written backwards gives 'nat'.
'g'+'nat'='GNAT'
(Other definitions for gnat that I've seen before include "Tang (anag.)" , "Any of various small biting flies" , "little irritation" , "small irritant" , "Insect often hovering over stagnant water" .)